The block matrix representations for the quasi-projection pairs on Hilbert $C^*$-modules

TL;DR

The paper addresses block-matrix representations for harmonious quasi-projection pairs $(P,Q)$ on Hilbert $C^*$-modules, where a harmonic pair requires polar decompositions of $P(I-Q)$ and $(I-P)Q$. It develops canonical $2\times2$ and Halmos-like $6\times6$ block representations, and derives new block forms for the associated projections $m(Q)$, $P_{\mathcal{R}(Q)}$, and $P_{\mathcal{N}(Q)}$, including when pairs are matched. Key contributions include explicit block formulas involving $A$, $\ell(A)$, and partial isometries, a supplementary projection $s(Q)$ with a new expression of $Q$ in terms of $m(Q)$ and $s(Q)$, and applications such as a new proof of the canonical form for quadratic operators and a demonstration that Krein-Krasnoselskii-Milman equality fails in general for quasi-projection pairs. These results advance operator-theoretic analysis on Hilbert $C^*$-modules and provide practical tools for studying projections, idempotents, and their distances within the $C^*$-algebra framework.

Abstract

A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored.

Theorems & Definitions (73)

• Lemma 2.1
• Lemma 2.2
• Remark 2.1
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.1
• Lemma 2.6
• Lemma 2.7
• Remark 2.2